1. Field of the Invention
This invention relates to nuclear measurements involving the spectroscopic analysis of energy spectra of gamma rays representative of atoms of elements under investigation, in order to determine the respective gamma ray contribution in percentage of different elements to a measured spectrum representative of an unknown material. From the percentage contributions (also called "elemental yields") may be derived concentration of these elements in the unknown material. By way of non limiting example, the invention can find application in nuclear well logging techniques, wherein a sonde is lowered in a well (or borehole) and carries out spectral measurements from which are derived information about the composition of the earth formation surrounding the borehole, or the borehole fluid, or the annulus including casing and cement located between the borehole wall and the formation. Alternately, examples of application for the present invention are material analysis using x-rays or gamma rays, medical analysis (by scanning) or airport security detection.
2. Related Art
A major goal of well logging is to obtain quantitative and qualitative information related to hydrocarbons in earth formation surrounding a well. In nuclear well logging, one carries out a spectral analysis of energy spectra of gamma rays resulting either from natural radioactivity or atom interactions of atoms with neutrons emitted from the sonde, such gamma rays being representative of certain atoms of the lithology or of the borehole fluid. Any reference hereafter made to "formation" or "lithology" has to be construed as referring to formation and/or borehole fluid. The invention cam also be applied to the nuclear tracer logging techniques.
For example, the energy spectrum of gamma rays resulting from the capture of thermal neutrons, after being decomposed into contributions due to individual atomic elements, usually called "elemental yields", reveals information concerning the presence of earth formation elements such as e.g. hydrogen, silicon, calcium, chlorine, sulfur and iron. Important petrophysical parameters such as porosity, matrix lithology and water salinity may be derived from the elemental yields. Examples of capture gamma ray spectra analysis are depicted in U.S. Pat. Nos. 3,521,064 to Moran et el., 4,464,569 to Flaum, 4,507,554 to Hertzog & Nelligan, 4,661,701 to Grau, 4,810,876 to Wraight et el.; 4,937,446 to Roscoe, Stoller and McKeon shows an inelastic gamma ray spectral analysis. All the above mentioned patents are assigned to the assignee of the present application, and are as well incorporated herein by reference. In the same vein, spectral analysis of natural gamma rays provides information on the uranium, thorium and potassium content of the earth formation, as shown e.g. in U.S. Pat. No. 3,976,878 to P. Chevalier & B. Seeman. As a further example, U.S. Pat. No. 4,166,216 to W. E. Cubberly shows a tracer logging method.
In accordance with the teaching of the above identified Moran Patent, a measured gamma ray energy spectrum, representative of a formation of unknown composition, is compared with a composite spectrum constructed from individual laboratory derived standard spectra of the constituents postulated to comprise the formation. The different amounts of the standard spectra (elemental yields) which give the best fit to the measured spectrum when weighted by each element sensitivity (i.e. the ability of an element to emit gamma rays) represent the relative proportion of the constituents of the formation. By appropriate selection of the standards, the proportion of the constituents of interest can be obtained and the desired information regarding hydrocarbon content may be derived.
The search for the best fit between the respective measured and composite spectra is, according to the method known so far, based on a linear least squares method, realizing a grid search for the minimum, such as depicted in the Moran Patent and in U.S. Pat. Nos. 3,928,763; 3,930,153 and 3,930,154 to H. D. Scott. Briefly stated, the least squares fitting method requires that the following function be minimum: ##EQU1## where "f.sub.k (E)" are the standard spectra; "Y.sub.k " are the corresponding elemental yields indicative of the proportion of that constituent "k" in the formation; "g(E)" is the function describing the measured spectrum, and "n" is the number of constituents, i.e. the number of standard spectra. Equation 1 could also be expressed in the matrix form: ##EQU2## where "U" represents the measured spectrum; "E.sup.i " is the difference between the composite spectrum and the measured spectrum, i.e. the error in the measured spectrum; "S.sub.k.sup.i " represents the standards, "Y.sub.k " are the unknowns i.e. the elemental yields, and "m" is the number of elemental energy intervals (or channels) of the spectrum. Minimizing the error in the fit means minimizing the chi-square "X.sup.2 ", in matrix notation: EQU X.sup.2 =(S Y-U).sup.T (S Y-U)=E.sup.T E (3)
The known linear least squares fitting method, although having been satisfactorily used up to now, shows some limitations, as explained hereafter.
The fit supposes that the respective measured and standard spectra have been obtained under equal or similar conditions. The parameters representative of these conditions are e.g. the gain, the offset, the background subtraction factor (hereafter referred to as BSF) and the resolution degradation factor (hereafter referred to as RDF). RDF is a parameter representative of the effects, on the detector measurements, of the variation from one detector to the other, or of the count rates effects due to the processing electronics downstream from the detector, or finally of the temperature which is of importance in the oil well logging application since the differences in temperature between the laboratory, where the standard spectra are obtained, and the Borehole environment, where the measured spectra are acquired, can be extreme.
These parameters are generally different from the measured spectrum to the composite spectrum. Thus, the measured spectrum must be corrected to match the hereabove mentioned parameters such as gain, offset, BSF of the composite spectrum, and the composite spectrum has to be corrected for RDF to match the resolution of the measured spectrum. This is done, in a known manner, by iteration through successive least squares fits of the measured spectrum to the composite spectrum while modifying one parameter at a time, the other parameters remaining constant. For example, the gain is modified during a first iterative process, the other parameters Being assigned a given value, until a fit is obtained. Then, a second iterative process is carried out until a fit is reached, by modifying the offset only, while the gain is assigned the optimum fitting value which has been determined during the first iteration, the other parameters remaining constant. This is followed by a process for causing the composite spectrum to be degraded in a manner which takes into account the effects of temperature on the detector resolution during the detection of the measured spectra, such as described in U.S. Pat. No. 4,394,579 to Grau and Hertzog and assigned to the assignee of the present application. A further iterative process is generally carried out for the background.
The known linear least squares fitting method as hereabove described involves the modification of one parameter at a time, i.e. the gain, then the offset, the RDF and finally the BSF, and is usually called a grid search.
This known method is not fully satisfactory since each parameter is actually correlated with the others. Thus, the value of one parameter which provides the best fit might not be the optimum value when the remaining parameters are given their own fitting values. In mathematical terms, this means that the fitting value of one (or several) of the parameters might have reached a local minimum instead of a global minimum, and thus might not represent the best fit. At best, even in assuming the global minimum has been reached, the method may converge very slowly towards the global minimum. One attempt to overcome this drawback has been to repeat and refine several times the above captioned iterative calculations, for each parameter. Nevertheless, this increases the complexity and the duration of the calculation without providing substantial benefit with regard to the uncertainty on the determination of the best fit.
Consequently, the need remains for a method providing a reliable fit between the measured spectrum and the composite spectrum, and thus improving the determination of the elemental yields.
Furthermore, the trend in the logging industry has always been to reduce the calculation time which allows one to make more measurements per time unit and/or use computer of reduced size.